The hiker needs to get from point A, to the river, then to the tent at point B. What is the shortest path?
Last year, at NCTM 2018, I attended a session run by the amazing Ron Lancaster
in which we solved the closely related "laying cable" problem.
Here, the cable is costlier to lay underwater than on land, so the goal is to minimize cost. Ron had us solve it about 9 different ways if I recall, using everything from algebra through calculus.
I can honestly say that I had neither of these consciously in mind (but obviously both had an impact on me) when, one day in September, I decided to have my geometry class work on what we took to calling "The Box Problem."
Barney the Ant wants to travel from point A to point B along the surface of the box. What is the shortest path?
We revisited this problem through the year, and solved it a few different ways. Here's what I did. (Note: the photos and data are from the work of different students working with different sized boxes, so they may not match each other.)
#1: Introducing the problem
There is a very simple solution to the problem, if you see it. (I won't give it away yet.) While one of my mantras this year (borrowed from David Butler) was "The End is Not the End," it would have been difficult to convince my students that we should keep working on a problem that we'd already solved. (Spoiler Alert: They bought into this approach eventually.) Thus, my first goal was to have them not see the quick way.
So: I got some actual boxes from our dining hall. When I introduced the problem in class, I used these to explain the question, thinking that perhaps it might be more difficult to see the easy answer on an actual box, rather than the diagram above. Plus, I could use my teacherly skills of distraction to manipulate the situation.
It worked. (One of the few times where I was rooting against a student having an insight.) We reduced the problem to the following question: To what point should Barney walk to on the top front edge in order to minimize distance. That is, find x in the diagram below.
#2: First approach: Data Gathering
Each group was given a box. They measured the dimensions of their box. Then, they took the top front edge of the box and divided it into 10 equal parts. Students drew the paths via each of the eleven points, then measured each distance using a meter stick.
From the table of data they made a guess as to what the shortest path is. Plotting the data in Desmos gave them a visual representation of the data and a possible solution.
#3: Second approach: The Pythagorean Theorem
Looking at the boxes, students recognized (through discussions) that they could compute the length of each path by using the Pythagorean theorem to find the hypotenuses of two triangles and add them together. Importantly, I had students create a table that showed the x-value, the equation used to find the computed distance via each x-value, the measured distance and the computed distance.
They graphed the computed data on Desmos, which made the estimated solution more obvious.
Both sets of data on the same graph led to a discussion about what the differences were and what might have caused them.
#4: Making some art
The next part of our exploration was taken directly from Ron Lancaster's workshop. The students used their boxes to cut pieces of string that were the actual length of each path they had drawn on the box. They tied one end of each string to a dowel, and hung a clothespin on the other end. Strings were spaced apart to match the divisions on the top edge of the box. This created a nice visual representation of the problem that we could hang on the wall.
Full disclosure: they initially cut the string by actually holding it along the path they had drawn on the box. This resulted in a little too much randomness (aka human error.) Instead, we redid the measurements using our calculated distances. This made for much cleaner and appealing art.
#5: Summarizing
I had students write a summary of all that we had done, in narrative form. I had them do this at this point in order to slow them down a little and let them reflect and consolidate what they'd done thus far.
#6: Graphing a function
I posed the question: What if you didn't know the actual value of x? What is the expression that would give the total distance? That is, can you create a formula?
It is with this question in mind that I'd had them show their work in the table column "equation used to find your distance." They had re-written it so many times that they very quickly saw the solution:
I'll pause here to point out how cool this is. In a regular geometry class, students created a function that represented the sum of two radicals, generated by them, from something they understood. Nice.
They graphed the equation on Desmos, and now were able to find the answer to the question (from a graph, at least.)
#7: Another approach - Similar Triangles
We put the box problem aside for a while, until we were studying similar triangles. At this point, I introduced the original shortest path problem:
We played with this for a bit, then we copied it onto patty paper. I had students find B', the reflection of point B in the line, and then they could see how to find the shortest path - travel directly from A to B'. To find the shortest path, we needed to solve a similar triangle problem.
At some point, a student said that it reminded them of the box problem. So we connected the two, and discovered that it's really the box problem in disguise. (Seeing the patty paper reflection was very helpful in making this clear.) If we fold up the top of the box, we can just draw the line between A and B, then solve a similar triangle problem. Here's my work for the box used in the Desmos graph above:
Very satisfying to see the same answer appear.
#8: A third approach: Trigonometry
Again, we put the box problem aside for a bit until we turned to right triangle trigonometry. I had students create a Geometer's Sketchpad diagram to model the box problem, with a free moving point along the middle line (top edge). They measured the angles in the middle:
The question: what is the relationship between the two angles measured at the point where the distance from A to B is the shortest? It wasn't hard to see that they would be equal -- we kind of knew that already from the similar triangle solution. But it means we can answer the question this way: Find the point F where angle DFA and angle BFC are equal. So going back to our diagram, we want to find what value of x makes y1 equal to y2.
We have two equations -- let's go to Desmos!
It's the same answer! This was very satisfying.
Notice, too, that geometry students are now graphing an arctan function that they created, with a purpose to solve a problem. Double cool.
#9: Epilogue: The Final Exam
Because we had become so familiar with this problem, I told them that it would be a large part of the final exam. After the standard part of they exam, they'd be able to use their computers if they wanted to type or use Desmos or Geometer's Sketchpad. They'd have plenty of time to work on it. Here's what I sent them before the exam. Their task was, essentially, to show me what they knew.
The results were great. To a student, they communicated something meaningful from the problem. Some only solved it using similar triangles, a few solved it in all the ways we solved it through the year. Some tried things we hadn't thought about.
I did do something either dumb or brilliant: In making up numbers, I accidentally made the opened up box a square. This allowed for some calculations to be done more easily. In fact, I was really pleased when some of the students recognized that their diagonal created an isosceles right triangle, which meant they knew the hypotenuse without using Pythagoras. Nice.
Also, the solution was an integer. Is this because the opened box was a square? Don't think so, but I'll have to think about that more -- but it points out again what a rich problem this is.
One student decided to try something new -- he looked at the area of the two triangles. His conclusion: "it didn't really help solve the problem, so never mind." Nice!
My favorite feedback: I asked one student about how difficult she found getting ready for the exam, and she told me, "At first, I was panicked about the box problem. But then when I realized I didn't have to memorize anything, I just had to understand it, I calmed down."
Here's one student's work (he also submitted a Desmos graph):
#10: My Reflections
I was teaching geometry for the first time in about 15 years, and I was kind of making up this problem as I went along. My experience tells me that often, making it up as you go along is the way to go. Too much structure might kill the spontaneity, and thus make the problem feel more routine. So, I will give my thoughts in the form of questions.
- Should I spray paint the boxes and have students draw nicer paths? The boxes sit around our classroom all year, so it might do to have them look good.
- Should I change from clothespins to something else? The way we did it this year, the sculptures were easily wrecked and didn't actually end up being displayed. The clothespins were nice, though, because they were easily adjustable.
- (Okay, I know the answer to this one.) I had students make a google doc for their initial response to the problem. I had meant to have them build on that document through the year, but I did not. Next year, I will.
- At some point while reading this, you might have thought, "Couldn't it be shorter if Barney went up the side of the box to the top, rather than the front?" This is a question I had hoped a student would raise at some point, but nobody did. I raised it late in the year. Should I raise it earlier?
- I had planned on having students potentially ask related questions, like: What if Barney is slower climbing up than climbing across? What could you deduce about the dimensions of the box if the solution to the problem was the midpoint of the top edge? How can we know quickly which route (front first or side first) is shorter? And so on. How can I work this in next year?
- Are there even more ways to solve the problem (that are accessible to geometry students.)?
- Is this a problem to repeat at all next year, or should I find another one?
My main takeaway is that I'm really pleased with the entire project. Initially, students found it strange that we were going to keep working on a problem we'd solved. But this is the generation that watches the same movie 140 times, so it wasn't much of a challenge to get them to think about the same thing over and over. On the final exam, instead of rushing through like I've usually seen, they were eager to show what they knew and even enjoyed the process. The work they did demonstrated (from my reading) good understanding of geometric concepts, good problem solving skills and most importantly, confidence in themselves as mathematicians.
Any suggestions or observations would be welcomed!
